Rodan: Difference between revisions

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'''Rodan''' is one of the notable [[extension]]s of the [[slendric]] [[regular temperament|temperament]], which divides [[3/2]] into three equal intervals representing [[8/7]] ([[tempering out]] the gamelisma, [[1029/1024]]), reaching the full [[7-limit]] such that 17 of these [[generators]] [[stacking|stack]] to reach the interval class of the [[5/1|5th harmonic]]. It tempers out [[245/243]], making it a [[sensamagic clan|sensamagic temperament]], so that [[5/3]] is divided into two intervals of [[9/7]]; and it tempers out [[5120/5103]], making it also a [[hemifamity temperaments|hemifamity temperament]], so that [[9/8]] stacks thrice into [[10/7]].

Unlike [[mothra]], which flattens the fifth to a [[meantone]] fifth, the fifth of rodan is slightly sharp of just, ranging from that of [[41edo]] to that of [[46edo]] (with [[87edo]] being an essentially optimal tuning). As a result, the [[256/243|diatonic minor second]] is compressed, and the interval known as the [[quark]], which represents [[49/48]], [[64/63]], and in rodan also [[81/80]], is even smaller than it is in tunings of slendric with a nearly just fifth. This entails that the [[~~MOS~~ scale]]s of rodan [[cluster MOS|cluster]] even more strongly around [[5edo]], although this can be thought of as an advantage in that it simplifies the conceptualization of rodan's inventory of intervals; rather than directly using ~~MOS~~ scales, which are either extremely imbalanced or overly large, an approach to rodan may involve picking and choosing which intervals from each [[pentatonic]] category to keep in the scale.

Unlike [[mothra]], which flattens the fifth to a [[meantone]] fifth, the fifth of rodan is slightly sharp of just, ranging from that of [[41edo]] to that of [[46edo]] (with [[87edo]] being an essentially optimal tuning). As a result, the [[256/243|diatonic minor second]] is compressed, and the interval known as the [[quark]], which represents [[49/48]], [[64/63]], and in rodan also [[81/80]], is even smaller than it is in tunings of slendric with a nearly just fifth. This entails that the [[mos scale]]s of rodan [[cluster MOS|cluster]] even more strongly around [[5edo]], although this can be thought of as an advantage in that it simplifies the conceptualization of rodan's inventory of intervals; rather than directly using mos scales, which are either extremely imbalanced or overly large, an approach to rodan may involve picking and choosing which intervals from each [[pentatonic]] category to keep in the scale.

As can be elucidated by [[S-expression]]s, rodan is very much a "counterpart" to mothra: the basic equivalence of slendric tempers S7 (49/48) = S8 (64/63), and mothra proceeds to equate it to S6 ([[36/35]]) as well; meanwhile, rodan extends the equivalence in the opposite direction to add S9 (81/80) to it, making it one of the five [[rank-2 temperament]]s definable by equating three adjacent square superparticulars.

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== Notation ==

A notation for rodan is listed in the notation guide for rank-2 pergens under [[pergen]] #8, {{nowrap|(P8, P5/3)}}. The generator is an upmajor ~~2nd~~. The [[enharmonic unisons in ups and downs notation|enharmonic unison]] is a ~~trudminor 2nd~~. Thus three ups equals a diatonic semitone, and three generators equals a perfect ~~5th~~. In rodan in particular, ^1 equals ~81/80 and ~64/63, and ^^1 equals ~33/32 and ~[[1053/1024]].

A notation for rodan is listed in the notation guide for rank-2 pergens under [[pergen]] #8, {{nowrap|(P8, P5/3)}}. The generator is an upmajor second. The [[enharmonic unisons in ups and downs notation|enharmonic unison]] is a triple-down minor second. Thus three ups equals a diatonic semitone, and three generators equals a perfect fifth. In rodan in particular, ^1 equals ~81/80 and ~64/63, and ^^1 equals ~33/32 and ~[[1053/1024]].

{| class="wikitable center-1 center-3"

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|-

| 5/4

| ~~Downmajor~~ third

| Down major third

| C−vE

|-

| 7/4

| ~~Downminor~~ seventh

| Down minor seventh

| C−vB♭

|-

| 11/8

| ~~Dup~~ fourth

| Double-up fourth

| C−^^F

|-

| 13/8

| ~~Dupminor~~ sixth

| Double-up minor sixth

| C−^^A♭

|}

Rodan's notation has much in common with that for 41edo and 46edo, since both edos map a minor ~~2nd~~ to three edosteps. It also resembles the notation for [[cassandra]]. All four notations notate the slendric tetrad (1–8/7–21/16–3/2) on C as C–^D–vF–G, and all four notations notate 5/4, 7/4, 11/8, and 13/8 as in the table above. But the notations diverge for other intervals, such as 11/10.

Rodan's notation has much in common with that for 41edo and 46edo, since both edos map a minor second to three edosteps. It also resembles the notation for [[cassandra]]. All four notations notate the slendric tetrad (1–8/7–21/16–3/2) on C as C–^D–vF–G, and all four notations notate 5/4, 7/4, 11/8, and 13/8 as in the table above. But the notations diverge for other intervals, such as 11/10.

== Chords ==

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== Music ==

; [[Gene Ward Smith]]

* ''Pianodactyl'' (archived 2010) – [https://soundcloud.com/genewardsmith/pianodactyl SoundCloud] | [http://www.archive.org/details/Pianodactyl detail] | [http://www.archive.org/download/Pianodactyl/pianodactyl.mp3 play] – ~~rodan~~[26] in 87edo tuning

* ''Pianodactyl'' (archived 2010) – [https://soundcloud.com/genewardsmith/pianodactyl SoundCloud] | [http://www.archive.org/details/Pianodactyl detail] | [http://www.archive.org/download/Pianodactyl/pianodactyl.mp3 play] – in Rodan[26], 87edo tuning

[[Category:Rodan| ]]

@@ Line 14: / Line 14: @@
 '''Rodan''' is one of the notable [[extension]]s of the [[slendric]] [[regular temperament|temperament]], which divides [[3/2]] into three equal intervals representing [[8/7]] ([[tempering out]] the gamelisma, [[1029/1024]]), reaching the full [[7-limit]] such that 17 of these [[generators]] [[stacking|stack]] to reach the interval class of the [[5/1|5th harmonic]]. It tempers out [[245/243]], making it a [[sensamagic clan|sensamagic temperament]], so that [[5/3]] is divided into two intervals of [[9/7]]; and it tempers out [[5120/5103]], making it also a [[hemifamity temperaments|hemifamity temperament]], so that [[9/8]] stacks thrice into [[10/7]].
-Unlike [[mothra]], which flattens the fifth to a [[meantone]] fifth, the fifth of rodan is slightly sharp of just, ranging from that of [[41edo]] to that of [[46edo]] (with [[87edo]] being an essentially optimal tuning). As a result, the [[256/243|diatonic minor second]] is compressed, and the interval known as the [[quark]], which represents [[49/48]], [[64/63]], and in rodan also [[81/80]], is even smaller than it is in tunings of slendric with a nearly just fifth. This entails that the [[MOS scale]]s of rodan [[cluster MOS|cluster]] even more strongly around [[5edo]], although this can be thought of as an advantage in that it simplifies the conceptualization of rodan's inventory of intervals; rather than directly using MOS scales, which are either extremely imbalanced or overly large, an approach to rodan may involve picking and choosing which intervals from each [[pentatonic]] category to keep in the scale.
+Unlike [[mothra]], which flattens the fifth to a [[meantone]] fifth, the fifth of rodan is slightly sharp of just, ranging from that of [[41edo]] to that of [[46edo]] (with [[87edo]] being an essentially optimal tuning). As a result, the [[256/243|diatonic minor second]] is compressed, and the interval known as the [[quark]], which represents [[49/48]], [[64/63]], and in rodan also [[81/80]], is even smaller than it is in tunings of slendric with a nearly just fifth. This entails that the [[mos scale]]s of rodan [[cluster MOS|cluster]] even more strongly around [[5edo]], although this can be thought of as an advantage in that it simplifies the conceptualization of rodan's inventory of intervals; rather than directly using mos scales, which are either extremely imbalanced or overly large, an approach to rodan may involve picking and choosing which intervals from each [[pentatonic]] category to keep in the scale.
 As can be elucidated by [[S-expression]]s, rodan is very much a "counterpart" to mothra: the basic equivalence of slendric tempers S7 (49/48) = S8 (64/63), and mothra proceeds to equate it to S6 ([[36/35]]) as well; meanwhile, rodan extends the equivalence in the opposite direction to add S9 (81/80) to it, making it one of the five [[rank-2 temperament]]s definable by equating three adjacent square superparticulars.
@@ Line 167: / Line 167: @@
 == Notation ==
-A notation for rodan is listed in the notation guide for rank-2 pergens under [[pergen]] #8, {{nowrap|(P8, P5/3)}}. The generator is an upmajor 2nd. The [[enharmonic unisons in ups and downs notation|enharmonic unison]] is a trudminor 2nd. Thus three ups equals a diatonic semitone, and three generators equals a perfect 5th. In rodan in particular, ^1 equals ~81/80 and ~64/63, and ^^1 equals ~33/32 and ~[[1053/1024]].
+A notation for rodan is listed in the notation guide for rank-2 pergens under [[pergen]] #8, {{nowrap|(P8, P5/3)}}. The generator is an upmajor second. The [[enharmonic unisons in ups and downs notation|enharmonic unison]] is a triple-down minor second. Thus three ups equals a diatonic semitone, and three generators equals a perfect fifth. In rodan in particular, ^1 equals ~81/80 and ~64/63, and ^^1 equals ~33/32 and ~[[1053/1024]].
 {| class="wikitable center-1 center-3"
@@ Line 181: / Line 181: @@
 |-
 | 5/4
-| Downmajor third
+| Down major third
 | C−vE
 |-
 | 7/4
-| Downminor seventh
+| Down minor seventh
 | C−vB♭
 |-
 | 11/8
-| Dup fourth
+| Double-up fourth
 | C−^^F
 |-
 | 13/8
-| Dupminor sixth
+| Double-up minor sixth
 | C−^^A♭
 |}
-Rodan's notation has much in common with that for 41edo and 46edo, since both edos map a minor 2nd to three edosteps. It also resembles the notation for [[cassandra]]. All four notations notate the slendric tetrad (1–8/7–21/16–3/2) on C as C–^D–vF–G, and all four notations notate 5/4, 7/4, 11/8, and 13/8 as in the table above. But the notations diverge for other intervals, such as 11/10.
+Rodan's notation has much in common with that for 41edo and 46edo, since both edos map a minor second to three edosteps. It also resembles the notation for [[cassandra]]. All four notations notate the slendric tetrad (1–8/7–21/16–3/2) on C as C–^D–vF–G, and all four notations notate 5/4, 7/4, 11/8, and 13/8 as in the table above. But the notations diverge for other intervals, such as 11/10.
 == Chords ==
@@ Line 397: / Line 397: @@
 == Music ==
 ; [[Gene Ward Smith]]
-* ''Pianodactyl'' (archived 2010) – [https://soundcloud.com/genewardsmith/pianodactyl SoundCloud] | [http://www.archive.org/details/Pianodactyl detail] | [http://www.archive.org/download/Pianodactyl/pianodactyl.mp3 play] – rodan[26] in 87edo tuning
+* ''Pianodactyl'' (archived 2010) – [https://soundcloud.com/genewardsmith/pianodactyl SoundCloud] | [http://www.archive.org/details/Pianodactyl detail] | [http://www.archive.org/download/Pianodactyl/pianodactyl.mp3 play] – in Rodan[26], 87edo tuning
 [[Category:Rodan| ]] <!-- main article -->